On Geometric Evolution and Microlocal Regularity of the Navier-Stokes Equations

We develop a geometric and microlocal framework for the Navier-Stokes equations by lifting the dynamics to the cosphere bundle of a Riemannian manifold. In this formulation, the velocity field and vorticity are represented as microlocal distributions whose evolution is governed by a linear transport-dissipation system generated by a canonical dynamical vector field. We introduce microlocal amplitudes, directional energy functionals, and monotone volume invariants on the compact phase space, which quantify directional concentration and alignment mechanisms associated with potential loss of regularity. The viscous term induces an effective geometric diffusion on the cosphere bundle, yielding closed differential inequalities in a geometric setting. To capture the interaction between fluid deformation and geometry, we define an effective connection and curvature tensor encoding the influence of the symmetric velocity gradient. This structure gives rise to a Ricci-type microlocal geometric evolution that constrains directional stretching and excludes extreme angular concentration compatible with viscous dissipation. While the present results do not resolve the global regularity problem, they provide a coherent geometric mechanism that severely restricts admissible blow--up scenarios, reformulating the regularity question as a problem of dissipative stability on a compact phase space.

Source: arXiv:2601.08854v2 - http://arxiv.org/abs/2601.08854v2 PDF: https://arxiv.org/pdf/2601.08854v2 Original Link: http://arxiv.org/abs/2601.08854v2